We are continuing to invent, develop, and translate novel Magnetic Resonance (MR) based water displacement imaging methods. Diffusion Tensor MRI (DT-MRI or DTI) is perhaps the best-known imaging method in this category that we invented and developed. It measures a diffusion tensor of mobile water within tissue. It consists of relating an effective diffusion tensor to the measured MR spin echo signal;estimating an effective diffusion tensor, D, in each pixel from a set of diffusion-weighted MR images;and calculating and displaying information derived from D. This information includes the local fiber-tract orientation, the mean-squared displacement water molecules migrate or diffuse in any given direction, the orientationally-averaged mean diffusivity, and other scalar invariant quantities that are independent of the laboratory coordinate system. These scalar parameters are intrinsic properties of the tissue and we measure them without contrast agents or dyes. For example, one DTI parameter, the orientationally-averaged diffusivity (or Trace), has been the most successful imaging parameter used to date to visualize an acute stroke in progress. Moreover, we have shown that DTI is effective in identifying Wallerian degeneration often associated with chronic stroke. Previous studies with kittens showed DTI to be useful in following early developmental changes occurring in cortical gray and white matter, which are not detectable using other means, which became the basis for applying these approaches in humans. The development of a method to color-encode nerve fiber orientation in the brain by Sinisa Pajevic and Carlo Pierpaoli has allowed us to identify and differentiate anatomical white matter pathways that have similar structure and composition, but different spatial orientations. The Direction-Encoded Color (DEC) maps of the human brain clearly show the main association, projection, and commissural white matter pathways, and are a mainstay in modern Neuroradiology practice. To assess anatomical connectivity between different functional regions in the brain, we also proposed and demonstrated a way to use DTI data to trace out nerve fiber tract trajectories, which we called DTI "tractography". This development was made possible by previous contributions of Sinisa Pajevic and Akram Aldroubi who implemented a general mathematical framework for obtaining a continuous, smooth approximation to the measured discrete, noisy, diffusion tensor field data. Collectively, these methods and approaches have allowed us and many other groups world-wide to perform detailed anatomical and structural analyses of the brain in vivo, which was only possible previously using laborious, invasive histological methods performed on dead tissue. Looking forward toward the use DTI in large, multi-center and multi-patient studies, we have been developing a variety of statistical techniques to be able to interpret our imaging data quantitatively, specifically to be able to determine the statistical significance of differences observed in our imaging data. To this end, we developed empirical Monte Carlo and Bootstrap methods for determining features of the statistical distribution of the diffusion tensor from experimental DTI data. Another innovation was a novel tensor-variate Gaussian distribution that fully describes the variability of the diffusion tensor in an idealized DTI experiment, and can be used to optimize the design and efficiency of DTI experiments. More recently, we developed approaches to measure uncertainties of many tensor-derived quantities, including the direction of nerve pathways. These collective developments provide the foundation for the use of powerful hypothesis tests to address a wide variety of important biological and clinical questions that previously could only be tackled using ad hoc methods, if at all. In the past few years, we have been developing sophisticated mathematiccal models of water diffusion profiles and related these to the MR signals we measure, with the aim of using MR data to infer new microstructural features of tissue (primarily white matter in the brain). One example of this idea was our composite hindered and restricted model of diffusion (CHARMED) MRI framework. A more recent enhancement of CHARMED, AxCaliber MRI, is another. AxCaliber allows us to measure the axon diameter distribution within a nerve bundle and the volume fraction of axons within that bundle from MR displacement imaging data. Sophisticated multiple pulsed field gradient NMR and MRI sequences, recently developed by Michael Komlosh, help us characterize microscopic anisotropy within tissues like gray matter that are macroscopically isotropic, appearing like a homogeneous and featureless gel in DTI. She and Ferenc Horkay have developed physical phantoms to test and interrogate our mathematical models of water diffusion in such complex tissues, being developed by Evren Ozarslan. He is also developing physical and mathematical models to relate other microstructural features of the underlying tissue to the MR signals we measure. Evren Ozarslan has been developing novel ways to interpret data obtained from the MR sequences to learn more about the size, shape and distribution of pores in biological tissue and other porous media. He has also used advanced mathematical techniques to characterize anomalous diffusion observed in various tissue specimen. Parameters derived from these novel measurements may provide a new source of MR contrast for promising neuroscience applications, such as in vivo Brodmann parcellation or clinical diagnostic applications, such as improved cancer detection and tumor staging. He has also developed novel approaches to characterize non-Gaussian features of the displacement distribution measured using MRI. Our group continues to work on reconstructing the average propagator (average displacement distribution) or features of it, using a relatively small number of diffusion weighted images (DWI). The average propagator is the "holy grail" of displacement imaging, which can by used to infer geometric features of microscopic restricted compartments as well as glean all of the information provided by DTI. Collectively, these methods represent a general framework for performing in vivo MRI histology--providing detailed microstructural and microarchitectural information that otherwise could only be obtained using laborious and invasive histological techniques applied on biopsied specimens or excised brains. We continue to develop new ways to assess tissue structure and architecture in vivo and non-invasively, with the aim of translating these approaches to the clinic, which we have done successfully with DTI, and to the larger research community.